Following [1]} with some improvements, we have three levels of confidence for spec-\(z\)  (ZCONF). ZCONF=1 is for a possible redshift with low confidence. It can be assigned due to low S/N values of lines (e.g., \(S/N\lesssim 3\) ), poor line fittings, existence of valid alternative redshift solutions, noisy narrow bands, or additional lines which present in the spectrum with reasonable S/Ns but can not be explained by the proposed redshift solutions. ZCONF=2 is for a good redshift. Regarding non-\(\mbox{Ly$\alpha $}\)  emitters (e.g., \(z\lesssim 2.9\) ), it is assigned for sources with multiple lines detected with good S/N values of \(S/N\gtrsim 5\) . For instance, a resolved [O II]\(\lambda \lambda \) 3726,3729 doublet with a clear narrow band would be sufficient for ZCONF=2. Regarding \(\mbox{Ly$\alpha $}\)  emitters, it is required to have a \(\mbox{Ly$\alpha $}\)  line with a good S/N of \(S/N\gtrsim 5\)  and a width and an asymmetry compatible with \(\mbox{Ly$\alpha $}\)  line shapes. The highest level, ZCONF=3, indicates a secure redshift. For non-\(\mbox{Ly$\alpha $}\)  emitters, the criteria are similar to those of ZCONF=2, but it is required to have more lines and higher S/N, together with high S/N narrow band images. For \(\mbox{Ly$\alpha $}\)  emitters, if there is no additional line except for \(\mbox{Ly$\alpha $}\) , it is required to have a \(\mbox{Ly$\alpha $}\)  line with a high S/N value of \(S/N\gtrsim 7\)  and a \(\mbox{Ly$\alpha $}\) -like line profile: a pronounced red asymmetrical line profile\(\mbox{Ly$\alpha $}\)  lines are fit with a skewed Gaussian with an asymmetry parameter \(\gamma \) : \(F(\lambda )=F_{\rm max}\left[1+ {\rm erf} \left( \gamma \frac{\lambda -\lambda _0}{\sqrt{2}\sigma }\right) \right] \left[ {\rm exp} \left( - \frac{(\lambda -\lambda _0)^2}{2\sigma ^2}\right) \right]\) , where \(F_{\rm max}\) , \(\sigma \) , and \(\lambda _0\)  represent the amplitude, the width, and the peak wavelength, respectively, and the erf is error function (see Bacon et al. in prep. for more details). (highly asymmetric of \(\gamma >2\) ) and/or a blue bump, or a double peaked line profile. The S/N thresholds are not strictly defined because other information was taken into account. In the case of an ORIGIN detection, and if the source can be matched to an HST counterpart, it adds confidence to the detection. In addition, if a photo-\(z\)  is reliable and a good match to the MUSE spec-\(z\) , it adds confidence to the redshift assignment. This could lead to a higher ZCONF than in the case of an ORIGIN source without HST counterpart. Meanwhile, faint HST sources often do not have a reliable photo-\(z\) , which can be in disagreement with a MUSE spec-\(z\) . It is not taken as a strong negative constraint (see Bacon et al. in prep.). In addition to the three ZCONF levels, ZCONF=0 is defined in the MXDF catalog (DR2 v0.8) for a source with an ORIGIN detection for which no spectroscopic redshift could be identified. However, in this paper, we independently defined ZCONF=0 as no spec-\(z\)  in the MXDF catalog with a tentative (best-estimate) assignment of a redshift or a constraint of a redshift.
